For that we need to derive the analytical expression of the scalar primordial spectrum beyond CMB scales, i.e., the expression of the “growing mode”. This would require the numerical integration of stochastic equations [1]}, [2]}, for different values of the model parameters (couplings and masses). However, the amplitude and tilt of the scalar spectrum mainly depend on the value of \(\kappa =d \ln \Upsilon /d \ln T\)  when the fluctuation leaves the horizon. Therefore instead of scanning over the model parameters, we will first derive the expression of the growing mode with \(\Upsilon \propto T^\kappa \) , for different values of constant \(\kappa \) . Now we can scan the model parameters, derive the value of \(\kappa \) , and then the spectrum at different stages during inflation: constraints on CMB scales set the parameter space consistent with observations, and for that we can explore whether it leads or not to enough amplification of GW, and PBHs. We will work with a quartic chaotic inflationary potential, still one the simplest inflation models, and consistent with observations when introducing dissipation in the system. The spectrum also depends on the statistical distribution function for inflaton fluctuations \(n_*\)  (whether in vacuum or thermal). We will focus on the vacuum case \(n_*=0\) . Although the thermal case can be also compatible with observations for the quartic chaotic model, typically it does not lead to enough amplification of the spectrum at the end.
